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Dec
29
revised Transcendental a infinitely close to rationals?
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Dec
29
comment Transcendental a infinitely close to rationals?
@GE, the OP upvoted my answer, so please don't put words in his mouth.
Dec
29
answered Is there a geometrical definition of a tangent line?
Dec
29
comment Transcendental a infinitely close to rationals?
@jwg, don't worry, Bishop Berkeley also thought infinitesimals were muddy, but that was because he missed the point of Leibnizian calculus; see ams.org/notices/201211/rtx121101550p.pdf
Dec
29
revised Transcendental a infinitely close to rationals?
edited tags
Dec
29
answered Transcendental a infinitely close to rationals?
Dec
25
comment How to add infinitesimal to the real number system
The question is why do we have $f(x+\epsilon)=f(x)+\epsilon f'(x)$. If this is your definition of $f(x+\epsilon)$ then you haven't accomplished anything. Also, the problem with the square root function is not merely at the origin. The problem is again that square root is not defined at points of the form $x+\epsilon$.
Dec
25
answered Why is “mathematical induction” called “mathematical”?
Dec
24
answered “Methods of Theoretical Physics for Mathematicians”
Dec
23
comment Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
That would be nice and Terry Tao has some ideas in this direction (he has a page on his blog called "cheap version of nonstandard analysis" or something like that). The problem is that as far as I can see the approach with the dual numbers does not work. I like the idea of finding cheap versions but Tao's is still considerably more involved than dual numbers. Again the difference is that Tao's approach works.
Dec
22
answered Difference between continuity and uniform continuity
Dec
22
comment Should a high school introductory calculus class teach $\varepsilon$-$\delta$ proofs?
I heard there is a trend in Italy to use nonstandard analysis in teaching calculus. In this approach one doesn't need epsilon-delta definitions when one defines the basic concepts of the calculus such as continuity and derivative. If you have any information about this please comment at math.stackexchange.com/questions/1057355/…
Dec
22
answered Understanding infinity
Dec
22
comment Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?
The correct technical term is "unlimited", see Goldblatt page 27.
Dec
22
comment Should a high school introductory calculus class teach $\varepsilon$-$\delta$ proofs?
@MathTeacher, you could consult this post: math.stackexchange.com/questions/1057355/…
Dec
22
answered Could “$\infty$” be understood by taking the reciprocals of the Hyperreal numbers?
Dec
22
comment Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
As you point out for polynomials one can do this algebraically but that's not very interesting or helpful, and does not seem to justify nilpotence at any rate.
Dec
22
comment Is the theory of dual numbers strong enough to develop real analysis, and does it resemble Newton's historical method for doing calculus?
you missed the point. As in Smooth Infinitesimal Analysis, one could define the derivative $L$ by $f(x+\epsilon)= f(x)+L\epsilon$. However, unlike Smooth Infinitesimal Analysis, there does not seem to be any useful way of defining $f$ at $x+\epsilon$ in the dual numbers.